Post on 01-Jun-2020
Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
Kazunobu Maruyoshi
Yukawa Institute for Theoretical Physics Kyoto University
based on KM and M. Taki, arXiv:1006.4505
July 6, 2010 @ KEK
Introduction
N=2 supersymmetric gauge theory is very interesting framework where various interesting results have been found: e.g.
• exact effective action related with the Seiberg-Witten curve
• instanton counting (Nekrasov partition function)
• relation with integrable systems
M5-branes wrapped on Riemann surface
N=2, SU(2) quiver SCFTs can be induced on worldvolume of 2 M5-branes wrapped on Riemann surfaces : [Witten ’97, Gaiotto ’09]
genus “genus” of quiver diagram
punctures flavor symmetries
complex structures gauge coupling constants
Low energy effective theory
• Seiberg-Witten curve is double cover of the Riemann surface on which M5-branes wrap:
• meromorphic one form is given by
Using these, the low energy prepotential is
SU(2) gauge theory with 4 flavors
A simple example: 2 M5-branes on , a sphere with 4 punctures induce SU(2) gauge theory with 4 flavors:
In this case, the curve obtained from M-theory is
Classical integrable systems
The relation between the Seiberg-Witten theory and the classical integrable system has been studied
[Gorsky et al., Donagi-Witten] [Martinec-Warner, Itoyama-Morozov, ….]
Seiberg-Witten curve spectral curve
meromorphic one form symplectic form
Pure Yang-Mills theory: periodic Toda systemN=2* gauge theory: elliptic Calogero-Moser system
Quantization of integrable systems[Nekrasov-Shatashvili]
N=2 low energy effective theory can be described in omega-background via Nekrasov partition function:
In the case where ,
related with the integrable system( plays the role of the Planck constant.)
Schrodinger equation and deformed prepotential [Mironov-Morozov, arXiv:0910.5670]
It was suggested that the deformed prepotential can be obtained from the periods
where P is calculated from the WKB ansatz
Relation with the Liouville theory[Alday-Gaiotto-Tachikawa]
N=2, SU(2) SCFT from M5-branes on
• Nekrasov partition function n-point conformal block
• mass parameters momenta of vertex operators
• vector multiplet scalar vev internal momenta
Liouville theory on
• deformation parameters
in central charge
Surface and loop operators from the Liouville
[Alday-Gaiotto-Gukov-Tachikawa-Verlinde]
The partition function in the presence of a surface operator was conjectured to be identified with
where is the degenerate field.
Loop operators correspond to monodromy operation:
Relation with quantum integrable system: Schrodinger equation
The first point is that the differential equation which is satisfied by the conformal block with degenerate field due to the null condition:
in the limit , can be identified with the Schrodinger equation of one-dimensional system
Relation with quantum integrable systems: deformed prepotential
The second point is that the proposal due to Mironov-Morozov is equivalent to the expected monodromy of the conformal block with the degenerate field, in the limit
Contents
1. Deformed prepotential from integrable system
2. AGT relation and its extension
3. Schrodinger equation from Liouville theory
4. Deformed prepotential from Liouville theory
5. Conclusion
1. Deformed prepotential from integrable system
Proposal by Mironov-Morozov
Let us consider the one-dimensional model:
where the Hamiltonian is .
The wave-function is given by
This is the exact WKB ansatz.
Proposal by Mironov-Morozov
Then, the deformed prepotential can be given by the similar relation to Seiberg-Witten theory:
This is very nontrivial proposal and there is no proof of this.
As we will see, we can check this at lower order in .
Prescription to obtain prepotential (1)
First of all, in terms of P, the Schrodinger equation is
Then, we obtain
which lead to
Prescription to obtain prepotential (2)
The contour integral can be written
Then,
1. calculate A- and B-cycle integrals at
2. act the operator on the ones obtained above
3. invert as and substitute it into
N=2* theory and elliptic Calogero-Moser
We consider the Hamiltonian:
which is the same as the one of the elliptic Calogero-Moser system.
The one form can be written Same form as the Seiberg-Witten curve of N=2* theory
Let be the ones obtained at
Then, at the next order,
By inverting the first equation and substituting it into the second, we obtain
This is the correct behavior of the Nekrasov function:
N=2 gauge theory with four flavors
The potential is
Then, the one form becomes
Similarly, we can obtain deformed prepotential.
2. AGT relation and its extension
Four-point conformal block
Let us consider the four-point correlation function which has the following form
The conformal block which we will consider behaves as
where B is expanded as
AGT relation (SU(2) theory with 4 flavors)
[Alday-Gaiotto-Tachikawa]
A simple example of the AGT relation is as follows:
the conformal block which we have seen above can be identified with the Nekrasov partition function of SU(2) theory with 4 flavors
× ×
× ×
Identification of the parameters
• External momenta : mass parameters
• Internal momenta : vector multiplet scalar vev
• (bare coupling) = (complex structure) = q
• deformation parameters ; Liouville parameter
AGT relation (N=2* gauge theory)
N=2*, SU(2) gauge theory is the theory with an adjoint hypermultiplet with mass m.
The partition function of this theory was identified with the torus one point conformal block:
where the identification is
Seiberg-Witten curve
The curve can be obtained by the insertion of the energy-momentum tensor:
and taking a limit where .
This limit corresponds to the gauge theory limit.
Seiberg-Witten curve (4 flavor theory)
In the case of a sphere with 4 punctures, the insertion leads to
Then, after some algebra, we obtainwith
Extension of the AGT relation
The partition function in the presence of the surface operator was conjectured to coincide with the conformal block with the degenerate field insertion:
where is the primary field with momentum . The degenerate field satisfies the null state condition:
Degenerate conformal block
We note that the degenerate conformal block may behave as
The second term indicates the expected behavior of the surface operator.
In what follows, we will consider the case with which would corresponds to the
quantum integrable system.
3. Schrodinger equation from Liouville theory
The case of a sphere with four punctures
The null state conditionin the case of a sphere with four punctures leads to the differential equation:
where
Let us go back to the gauge theory parameters:
In this scaling, e.g.
Then, by taking the limit where ,
where V is the same as the potential above.
Note that the limit is crucial to obtain the form of the Schrodinger equation.
The case of a torus with one puncture
We will skip the detailed derivation of the differential equation. The result is
Again, we go back to the gauge theory parameters and then take the limit
This is the Hamiltonian of the elliptic Calogero-Moser system.
4. Deformed prepotential from Liouville theory
Differential equation
Let us solve the differential equation obtained above:
where g(z) is an irrelevant factor which depends on z.
Note that the difference between this equation and the Schrodinger equation is
WKB ansatz
Let us make an ansatz:
Then, the differential equation leads to
which implies
where
Contour integrals
At lower order, the contour integral of one form is
If we assume the proposal:
we obtain
Thus,
Here we come to the important relation.
The final remark is the following. We already know the form of u: indeed we can show that
Therefore, we obtain
Expected monodromy
From the Liouville side, monodromy of has been considered: [Alday et al., Drukker-Gomis-Okuda-Teschner]
In the limit where , these become
These are exactly what we have derived!
5. Conclusion
We have considered the proposal that the deformed prepotential can be obtained from the Schrodinger equation.
We have derived the Schrodinger equation from the conformal block with the degenerate field by making use of the AGT relation
We have seen that the proposal above is equivalent to the expected monodromy of the conformal block.
Future directions
Higher order check and other SU(2) theories
SU(N)/Toda generalization: Surface operators and degenerate field insertion
Differential equation from the matrix model by Dijkgraaf-Vafa
Direct check that the Nekrasov partition function with surface operator satisfies the differential equation
Deformation to N=1 ?
Thank you very much for your attention!